Cryptographic method protected against covert channel type attacks

ABSTRACT

The invention relates to a cryptographic method secured against a covert channel attack. According to the invention, in order to carry out a selected block of instructions as a function of an input variable amongst N predefined instruction blocks, a common block is carried out on the predefined N instruction blocks, a predefined number of times, the predefined number being associated with the selected instruction block.

The invention relates to a cryptographic method protected against attacks of the covert channel type. The invention is in particular advantageous for protecting algorithms during which a block of instructions from amongst several different blocks of instructions is executed as a function of an input variable. Such an algorithm is for example, but not limitingly, a binary exponentiation algorithm performing a calculation of the type B=A^(D), with A, B and D being integer numbers. Such an algorithm is for example implemented in electronic devices such as chip cards.

The outline diagram of such an algorithm is depicted in FIG. 1. It comprises a first step of testing the value of an input data item. According to the result of the test, a block of instructions Π₀ or a block of instructions Π₁ is carried out. The algorithm can then terminate, or a new test step is performed on another input variable. In the example of an operation of the type B=A^(D), the input variable is a bit D_(i) of D and the diagram in FIG. 1 is repeated successively for each bit of D.

The blocks of instructions Π₀, Π₁ each comprise a set of instructions to be executed, for example operations of addition, multiplication, variable updating, etc. The number and/or the type of instruction may be different from one block of instructions Π₀, Π₁ to the other.

Many cryptographic algorithms are based on the outline diagram in FIG. 1. This is in particular the case with cryptographic algorithms based on exponentiation calculations of the type B=A^(D), where A, B are integer numbers usually of large size, and D a predetermined number of M bits.

The numbers A, B may correspond for example to a text which is enciphered or to be enciphered, a data item which is signed or to be signed, a data item which is verified or to be verified, etc. The number D may correspond to elements of keys, private or public, used for enciphering or deciphering the numbers A, B.

By way of example of the algorithms such as the so-called “Square-And-Multiply” algorithm, the so-called “Right-To-Left binary algorithm” and the so-called “(M, M³) algorithm” may be used for performing exponentiation calculations.

A malevolent user may possibly undertake attacks aimed at discovering in particular confidential information (such as for example the key D or a data item derived from this key) manipulated in processings carried out by the calculation device executing an exponentiation operation.

A simple attack, known as a “timing attack”, against the algorithm in FIG. 1 consists in measuring the time necessary for the device to execute a block of instructions between two test steps. If the execution times for the blocks of instructions Π₀, Π₁ are different, then it is easy to identify a block of instructions Π₀ or Π₁ and to deduce therefrom the value of the associated input variable.

In order to protect against this attack, it is possible to add fictional instructions in the shortest block of instructions Π₀ or Π₁ (a block of instructions is “the shortest” if the time taken to perform it is the least) so that the two blocks of instructions Π₀, Π₁ are of the same duration.

. . . in question, according to the value of this bit and/or according to the instruction.

Covert channel attacks may succeed with algorithms such as the one in FIG. 1 if the blocks of instructions Π₀, Π₁ are not equivalent vis-a-vis these attacks.

The term “equivalent” must be understood here and throughout the remainder of the text in the following manner. Two instructions INST₁, INST₂ (or two blocks of instructions Π₀, Π₁) are said to be equivalent (INST₀ is denoted˜INST₁) if it is not possible to differentiate them by means of a covert channel attack. This is the case in particular if the physical quantity measured during the attack follows the same development for the two instructions. It should be noted however that two instructions may be equivalent vis-à-vis one covert channel attack and not be equivalent vis-à-vis another covert channel attack.

In the same way, it will be said that two instructions (or blocks of instructions) are equal if, when they are used with the same input data, they produce identical output data.

It is known how to protect against covert channel attacks by adding fictional instructions to the algorithm. It is assumed hereinafter that a fictional instruction is equivalent to a similar real instruction. For example, the instruction i<-i−0 is assumed to be equivalent to the instruction i<-i−1.

In the case of the algorithm in FIG. 1, it is thus known how to effect a fictional block of instructions Π₁ after each block of instructions Π₀, and to effect in a symmetrical manner a fictional block of instructions Π₀ before each block of instructions Π₁ (see the steps in dotted lines in FIG. 1). Thus, whatever the value of the input data item, a block of instructions Π₀ and a block of instructions Π₁ will be effected, in this order, one or other being fictional, so that it is not possible to predict the value of the input data item, the physical quantities relating to a calculation being equivalent. Thus there is denoted: (Π₀||Π_(1(fictional)))˜(Π_(0(fictional))||Π₁).

The notation “||” signifies the successive effecting of blocks of instructions Π₀, Π₁ (or more generally the successive effecting of two instructions).

Though this solution is effective against covert channel attacks, it does however have the drawback of multiplying on average by two the time needed for executing the algorithm.

This is because, in the case of an unprotected algorithm using M input data (for example the M bits of a data item D), statistically on average M/2 blocks of instructions Π₀ and M/2 blocks of instructions Π₁ are effected. If T0 and respectively T1 are the average times for executing a block of instructions Π₀ or respectively Π₁, then the average time for executing the unprotected algorithm is equal to M*(T0+T1)/2.

On the other hand, in the case of the algorithm protected by fictional blocks of instructions Π₀, Π₁, a block of instructions Π₀ and a block of instructions Π₁ are systematically effected for each of the M input data. Consequently the average time for executing the algorithm protected by fictional blocks of instructions is equal to M*(T0+T1).

A first aim of the invention is to propose a novel cryptographic algorithm protected against covert channel attacks. A second aim of the invention is a protected cryptographic method which is more rapid than existing protected algorithms.

This aim is achieved by a cryptographic calculation method according to the invention, characterised in that, in order to execute a chosen block of instructions (Π_(j)) as a function of an input variable (D_(i)) from amongst N predefined blocks of instructions (Π₁, . . . , Π_(N)), a block (Γ(k,s)) common to the N predefined blocks of instructions (Π₁, . . . , Π_(N)) is executed a predefined number of times (L_(j)), the predefined number (L_(j)) being associated with the chosen block of instructions (Π_(j)).

In other words, according to the invention, a single elementary block, the common elementary block, is effected whatever the input variable. The common elementary block is executed a predefined number of times, according to the input variable. Contrary to the known methods, different blocks of instructions are not executed as a function of the input variable.

Thus, with the invention, it is then not possible to determine, by means of a covert channel attack, which block of instructions is executed. A method according to the invention is therefore well protected.

The predefined number (L_(j)) is variable from one predefined block of instructions (Π₁, . . . , Π_(N)) to another.

The common block (Γ(k,s)) preferably comprises at least one calculation instruction (γ(k,s)) equivalent vis-à-vis a covert channel attack to a calculation instruction for each predefined block (Π₁, . . . , Π_(N)).

The common block (Γ(k,s)) can also comprise an instruction to update a loop pointer (k) indicating a number of executions already executed of the common block (Γ(k,s)).

If necessary, the common block (Γ(k,s)) can additionally comprise an instruction to update a state pointer (s) indicating whether the predefined number (L_(j)) has been reached.

The value of the loop pointer (k) and/or the value of the state pointer (s) are a function of the value of the input variable (D_(i)) and/or of the number of instructions of the instruction block (Π_(j)) associated with the value of the input variable.

Preferably, if several common blocks are possible, the common block is chosen so as to be minimum, in the sense that it comprises a minimum number of instructions and/or in that it is effected in a minimum time.

Preferably again, in order to successively effect several blocks of instructions chosen from amongst the N predefined blocks of instructions (Π₁, . . . , Π_(N)), each chosen block of instructions being selected as a function of an input variable (Di) associated with an input index (i),

the common block (Γ(k,s)) is executed a total number (L_(T)) of times, the total number (L_(T)) being equal to a sum of the predefined numbers (L_(j)) associated with each chosen block of instructions (Π_(j)).

There too, in order to successively execute several blocks of instructions, only the common block is executed an appropriate number of times; this whatever the blocks of instructions to be executed. It is therefore not possible to predict with which block of instructions the common block currently being executed is associated. A covert channel attack can therefore not succeed.

It should be noted that one and the same block of instructions (Π_(j)) can be chosen several times according to the input variable (Di) associated with the input index (i).

According to one embodiment of the invention, one or more mathematical relationships are used in order to update the loop pointer and/or the state pointer and/or indices of registers used for implementing the cryptographic method and/or the input variable or variables. According to another embodiment of the invention, the updating takes place using a table with several inputs. These embodiments will be detailed at greater length hereinafter by means of practical examples.

The invention also relates to a method for obtaining a block (Γ(k,s)) common to N predefined blocks of instructions (Π₁, . . . , Π_(N)). The said method is able to be used for implementing a protected cryptographic calculation method according to the invention such as the one described above.

According to the invention, a common block (Γ(k,s)) is obtained by performing the following steps:

E1: breaking down each predefined block of instructions (Π₁, . . . , Π_(N)) into a series of elementary blocks (γ) equivalent vis-à-vis a covert channel attack, and classifying all the elementary blocks (for example by allocating a rank),

E2: seeking a common elementary block (γ(k,s)) equivalent to all the elementary blocks (γ) of all the predefined blocks of instructions,

E3: seeking a common block (Γ(k,s)) comprising at least the common elementary block (γ(k,s)) previously obtained during step E2 and an instruction to update a loop pointer (k) such that an execution of the common elementary block associated with the value of the loop pointer (k) and an execution of the elementary block with a rank equal to the value of the loop pointer (k) are identical.

If necessary, during step E1, one or more fictional instructions can be added to the series of instructions of one or more blocks of instructions. This can facilitate the breaking down of each block of instructions into elementary blocks all equivalent vis-à-vis a covert channel attack.

During step E1, each predefined block of instructions Π₁ to Π_(N) is divided into elementary blocks which are equivalent vis-à-vis a given attack; the elementary blocks are classified. For example: Π₁=γ1||γ2||γ3; Π₂=γ4||γ5; . . .

More generally, each block of instructions Π₁, . . . , Π_(N) is broken down thus: Π₁=γ(C ₁)|| . . . ||γ(C ₁ +L ₁−1) Π₂=γ(c ₂)|| . . . ||γ(c ₂ +L ₂−1) . . . Π_(j)=γ(c _(j))|| . . . ||γ(c _(j) +L _(j)−1), . . . Π_(N)=γ(c _(N))|| . . . ||γ(c _(N) +L _(N)−1) with C₁=0 C ₂ =L ₁ . . . C _(j) =L ₁ +L ₂ + . . . +L _(j−1) . . . C _(N) =L ₁ + . . . L _(N−1)

L_(j) is the number of elementary blocks necessary for completely breaking down the predefined block of instructions Π_(j).

During step E2, a common elementary block γ is sought such that each block of instructions Π_(J)(1≦j≦N) can be expressed in the form of a repetition L_(j) times of the common elementary block γ.

The common block is preferably chosen so as to be minimum. In other words, it comprises a minimum number of instructions and/or is executed in a minimum time.

During step E3, a common block is sought comprising:

-   -   one or more common elementary blocks obtained during step E2,         and     -   an instruction to update a loop pointer (k) such that an         execution of the common elementary block associated with the         value of the loop pointer (k) and an execution of the elementary         block with a rank equal to the value of the loop pointer (k) are         identical.

If necessary, a state pointer s can also be used in addition to the loop pointer:

-   -   the state pointer s indicates whether the common elementary         block has already been executed a predefined number of times         corresponding to the number L_(j) of elementary blocks breaking         down a given block of instructions Π_(j); in one example, the         state pointer s is equal to 1 when the predefined number L_(j)         of elementary blocks has been executed, and is equal to 0         otherwise;     -   the loop pointer indicates the rank of the elementary block to         be executed amongst all the elementary blocks. In very general         terms, the loop pointer can be defined in all cases according to         the following Equation 1:         k<-(/s)·(k+1)+s·f(D _(i))

D_(i) is the input variable for selecting a block of instructions to be executed, s is the state pointer, and f is a logic function of the input variable D_(i) associated with a predefined block of instructions Π_(j) to be executed, and /s is the complement of the pointer s (logic NOT function).

The above equation giving the value k is obtained by means of the following reasoning.

When a block of instructions Π_(j) is effected, the loop pointer k must be incremented by 1 at each execution of the common elementary block (associated with an equivalent elementary block of the breaking down of the block Π_(j)) as long as s=0, that is to say as long as the number of elementary blocks associated with the block Π_(j) has not been reached. This is represented by the instruction: k<-(k+1)

-   -   when s=0

Conversely, when the common elementary block associated with the last elementary block of the block Π_(j) (that is to say when s=1) is effected, it is necessary to modify k so as to effect the common elementary block associated with the first elementary block of the following block of instructions Π_(j′). This results in the following instruction: k<-f(D _(i))

-   -   when s=1

where D_(i) is the input variable which determines the choice of the calculation Π_(j), to be effected.

By combining the last two instructions, Equation 1 is finally obtained.

The above equation giving the value of k as a function of s is valid in all cases. In certain particular cases, this equation may be modified as will be seen better below in practical examples.

The invention and the advantages which stem from it will appear more clearly from a reading of the following description of examples of implementation of a protected cryptographic method according to the invention. The description is to be read with reference to the accompanying drawings, in which:

FIG. 1 is a generic diagram of known methods able to be protected according to the invention,

FIG. 2 is a diagram of the generic method of FIG. 1 protected according to the invention,

FIGS. 3 and 4 detail the implementation of certain steps of the method of FIG. 2 in the case of known exponentiation methods, protected according to the invention.

In the examples which follow, the obtaining of a common elementary block according to the invention and the use of this elementary block will be described in particular, in the practical cases of cryptographic calculation methods.

EXAMPLE 1

In a first practical example, an exponentiation algorithm of the “Square-and-Multiply” type is considered, which makes it possible to perform an exponentiation operation of the type B=A^(D), where D=(D_(M-1), . . . , D₀) is a number of Mbits. The known form of this algorithm can be represented as follows:

Initialisation: R₀<-1; R₁<-A; i<-M−1

As long as i≦0, repeat:

-   -   If D_(i)=1, then effect Π₀:         R ₀<-R ₀ ×R ₀         R ₀<-R ₀ ×R ₁         i<-i−1     -   If D_(i)=0, then effect Π₁:         R ₀<-R ₀ ×R ₀         i<-i−1

Return R₀.

Algorithm 1 Non-Protected “Square-and-Multiply”

R₀, R₁ are registers of a calculation device adapted for implementing the algorithm, and i is a loop index referencing the various bits of D. According to the value D_(i), Π_(j)=Π₀ or Π_(j)=Π₁ is executed.

In Algorithm 1, the blocks of instructions Π₀, Π₁ are effected according to the value of a bit D_(i) of the exponent D, and the loop index i is decremented at the end of each block of instructions Π₀, Π₁ so as to successively process all the bits D_(i) of D.

In Algorithm 1, the blocks of instructions Π₀, Π₁ are not equivalent vis-à-vis a covert channel attack, in particular because the number of instructions of Π₀ is different from the number of instructions of Π₁.

In order to protect Algorithm 1 according to the invention, a common elementary block Γ able to be used for executing the blocks Π₀, Π₁ is sought.

For this purpose, each block of instructions Π₀, Π₁ is first of all broken down into a series of elementary blocks, all equivalent to each other vis-à-vis a given attack.

The block of instructions Π₀ can be written: R ₀<-R ₀ ×R ₀ i<-i−0 R ₀<-R ₀ ×R ₁ i<-i−1

The instruction i<-i−0 is fictional: it does not modify any variable, any data item manipulated by Algorithm 1.

Π₀ can then be broken down into two elementary blocks:

Π₀=γ₀||γ1 with

-   -   γ0: R₀<-R₀×R₀         -   i<-i−0     -   γ1: R₀<-R₀×R₁         -   i<-i−1

Π₁ is broken down in the same way into an elementary block:

Π₁=γ2 with

-   -   γ2: R₀<-R₀×R₀         -   i<-i−1

It should be noted that the blocks γ0, γ1, γ2 are all equivalent (γ0˜γ1˜γ2) vis-à-vis a covert channel attack if it is assumed that the instructions R₀<-R₀×R₀ and R₀<-R₀×R₁ are equivalent and that the instructions i<-i−0 and i<-i−1 are equivalent.

Thus each block of instructions Π₀, Π₁ has been broken down into a variable number of elementary blocks (variable from one block of instructions to another), all equivalent to each other.

Next a state pointer s and a rank pointer k are defined. When a block of instructions Π_(j) is in the course of execution:

-   -   k is used to indicate which elementary block γk is to be         effected; the value of k depends in particular on the block         Π_(j) currently being executed (and therefore on the input         variable D_(i) tested) and the state of advancement of the         execution of the block Π_(j)     -   s is used to indicate whether at least one elementary block γk         is yet to be effected or whether the current block γk is the         last of the block of instructions Π_(j).

In the case of the above example relating to Algorithm 1, the development of the pointers k, s can be summarised by the following table. TABLE 1 k s (D_(i) = 1) γ0: R₀ <- R₀ × R₀; i <- i − 0 0 0 γ1: R₀ <- R₀ × R₁; i <- i − 1 1 1 (D_(i) = 0) γ2: R₀ <- R₀ × R₀; i <- i − 1 2 1

s can be calculated from k: if the elementary block γk which is to be effected is the last elementary block of a block Π, then s=1, otherwise s=0.

In the case of Algorithm 1, it is possible for example to calculate s by means of the following equation: s=(k mod 2)+(k div 2)   (EQ a)

“div” designates an integer division and “mod” a modular reduction. From Equation 1, the various values of s as a function of k are found (cf Table 1).

The updating of k can be obtained from s and D_(i), D_(i) representing the current block Π_(j):

-   -   if s=0 (block Π_(j) currently being effected), k is incremented         by 1 at each effecting of an elementary block γ, in order then         to effect the following elementary block γ.     -   if s=1, the current block Π is terminated and the following         elementary block γ to be effected is the first elementary block         of the next block Π_(j) to be executed; k therefore depends on         D_(i).

From the above, it is deduced therefrom that k can be obtained by the following relationship: k<-(/s)×(k+1 )+s×f(D _(i))   (EQ b)

/s is the complementary value of s (logic NOT function), and f is a logic function of D_(i), which depends on the algorithm to be protected (see also FIG. 3).

In the case of Algorithm 1, it is possible for example to choose f(D_(i))=2×(/D_(i)).

Thus, with Equation 3: k<-(/s)×(k+1 )+s×2×(/D _(i))   (EQ c)

the various values of k are found as a function of s and D_(i) (cf Table 1).

Finally, a common elementary block γ(k,s), is defined, equivalent to the elementary blocks γ0, γ1, γ2 and such that γ(0, 0)=γ0, γ(1, 1)=γ1 and γ(2, 1)=γ2.

For Algorithm 1, it is possible for example to choose:

γ(k,s): R₀<-R₀×R_(k mod 2)

-   -   -   i<-i−s

Using the common elementary block γ(k,s), Algorithm 1 can finally be written (see also FIG. 3):

Initialisation: R₀<-1; R₁<-A; i<-M−1

As long as i≧0, repeat the common block Γ(k,s): k<-(/s)×(k+1)+s×2×(/D _(i)) s<-(k mod 2)+(k div 2)

-   -   γ(k,s): R₀<-R₀×R_(k mod) 2         -   i<-i−s

Return R₀.

Protected Algorithm 1

(Protected “Square-and-Multiply” Algorithm)

In this algorithm, a single common block Γ(k,s) is used, whatever the values of D_(i). In other words, whatever the value of D_(i), the same instruction or the same block of instructions is executed. In the case where D_(i)=0, the block Γ(k,s) is executed only once. In the case where D_(i)=1, the common block Γ(k,s) is executed successively twice.

Whatever the values of the pointers k, s and whatever the value of D_(i), the associated block Γ(k,s) is equivalent, vis-à-vis a covert channel attack, to the block Γ(k,s) previously executed and to the block Γ(k,s) next executed. Consequently it is not possible to distinguish them from each other and it is not possible to know to which block of instructions Π_(j) the common block Γ(k,s) currently being executed corresponds.

It should be noted that, with respect to the non-protected Algorithm 1, the protected Algorithm 1 according to the invention uses the same number of calculation instructions (such as multiplication instructions for example) in order to arrive at the same final result. The protected Algorithm 1 according to the invention simply comprises additional steps of updating pointers: such steps are much more rapid and consume much fewer resources than a calculation instruction such as a multiplication. Consequently the time for executing the protected algorithm is almost the same as that of the non-protected Algorithm 1: Tex=1.5*M*T0, T0 being the time for executing a multiplication.

It should also be noted that the common block Γ(k,s) is not unique for one and the same algorithm, as will be seen with Example 2.

EXAMPLE 2

In the case of the “Square and Multiply” algorithm, other breakdowns of the block of instructions Π₀ can be envisaged, for example:

Π₀=γ′0||γ′1 with

-   -   γ′0: R₀>-R₀×R₀         -   i<-i−1     -   γ′1: R₀<-R₀×R₁         -   i<-i−0

This breakdown can be envisaged since the fictional instruction i<-i−0 can be executed at any time during the block Π₀. It is consequently found that the elementary blocks γ′0 and γ2 are identical. Table 1 is then modified in the following manner. TABLE 2 k s (D_(i) = 1) γ′0: R₀ <- R₀ × R₀; i <- i − 1 0 0 γ′1: R₀ <- R₀ × R₁; i <- i − 0 1 1 (D_(i) = 0) γ′0: R₀ <- R₀ × R₀; i <- i − 1 0 1

The pointer s here becomes superfluous since only two elementary blocks are possible, γ′0 and γ′1. Finally, the common elementary block γ′(k,s) and the following protected algorithm are obtained (see also FIG. 4):

Initialisation: R₀<-1; R₁<-A; i<-M−1; k<-1

As long as i≧0, repeat the common block Γ′(k,s): k<-(D _(i)) AND (/k)

-   -   γ′(s,k): R₀<-R₀×R_(k)         -   i<-i−(/k)

Return R₀.

Protected Algorithm 2

(Protected “Square-And-Multiply” Algorithm, Version 2)

EXAMPLE 3

The exponentiation algorithm known as the “Right-To-Left binary algorithm” is fairly similar to the “Square-And-Multiply” algorithm. It makes it possible to perform an operation of the type B=A^(D), starting from the least significant bit of D in the following manner:

Initialisation: R₀<-1; R₁<-A; i<-0

As long as i≦M−1, repeat:

-   -   If D_(i)=1, then effect the block Π₀:         R ₀<-R ₀ ×R ₁         R ₁<-R ₁ ×R ₁         i<-i+1     -   If D_(i)=0, then effect the block Π₁:         R ₁<-R ₁ ×R ₁         i<-i+1

Return R₀.

So-called “Right-To-Left Binary Algorithm”

The blocks Π₀, Π₁ in this example can be broken down in the following manner: TABLE 3 k s Π₀ (D_(i) = 1) γ0: R₀ <- R₀ × R₁; i <- i + 0 0 0 γ1: R₁ <- R₁ × R₁; i <- i + 1 1 1 Π₁ (D_(i) = 0) γ0: R₁ <- R₁ × R₁; i <- i + 1 0 1

Here also, as only two elementary blocks γ0, γ1 are used to break down Π₀, Π₁, the pointer s is unnecessary. It is possible for example to choose the following common elementary block γ(k):

-   -   γ(k): R_(k)<-R_(k)×R₁         -   i<-i+k

and to update the pointer k before each effecting of the block γ(k) using the instruction k<-k ⊕ D_(i), where ⊕ designates the exclusive-OR operator (⊕) Finally the following protected Algorithm 3 is obtained:

Initialisation: R₀<-1; R₁<-A; i<-0; k<-1

As long as i≦M-1, repeat the block Γ(k,s): k<-k ⊕ D _(i)

-   -   γ(k): R_(k)<-R_(k)×R₁         -   i<-i+k

Return R₀.

Algorithm 3

(Protected “Right-To-Left Binary Algorithm”)

The above examples describe algorithms during which only two blocks of instructions Π₀ or Π₁ are executed as a function of the value of an input variable D_(i). The invention can however apply to algorithms using more than two blocks of instructions Π.

EXAMPLE 4

In this example the so-called “(M, M³) algorithm” is considered, known in the following form:

Initialisation: R₀<-1; R₁<-A; R₂<-A³; D₋₁<-0; i<-M−1

As long as i≧0, repeat:

-   -   If Di=0, effect Π₀:         R ₀<-(R₀)²         i<-i−1     -   If D_(i)=1 AND (D_(i-1)=0), effect Π₁:         R ₀<-(R ₀)²         R ₀<-R ₀ ×R ₁         i<-i−1     -   If D_(i)=1 AND (D_(i−1)=1) , effect Π₂:         R ₀<-(R ₀)²         R ₀<-(R ₀)²         R ₀<-R ₀ ×R ₂         i<-i−2

Return R₀.

So-Called “(M, M³) Algorithm”

AND is the logic AND function. R₀, R₁, R₂ are registers of the device used for implementing the algorithm.

By replacing the (R₀)² type squares with R₀×R₀ type multiplications, and introducing fictional instructions of the type i<-i−0, it is possible to break down the algorithm (M, M³) according to the table: TABLE 4 k s Π₀ (D_(i) = 0) γ0: R₀ <- R₀ × R₀; i <- i − 1 0 1 Π₁ (D_(i) = 1) and (D_(i−1) = 0) γ1: R₀ <- R₀ × R₀; i <- i − 0 1 0 γ2: R₀ <- R₀ × R₁; i <- i − 1 2 1 Π_(s) (D_(i) = 1) and (D_(i−1) = 1) γ3: R₀ <- R₀ × R₀; i <- i − 0 3 0 γ4: R₀ <- R₀ × R₀; i <- i − 0 4 0 γ5: R₀ <- R₀ × R₂; i <- i − 2 5 1

Table 4 makes it possible to fairly easily calculate the value of the pointer k as a function of s and D_(i), and the value of the pointer s as a function of k, as before. Moreover, the blocks γ0 to γ5 are all equivalent vis-à-vis a covert channel attack, and it is possible for example to choose the following common elementary block γ(k,s):

-   -   γ(k,s): R₀<-R₀×R_(s×(k div 2))         -   i<-i−s×(k mod 2+1)

Finally, a protected Algorithm 4 is derived from this:

Initialisation: R₀<-1; R₁<-A,; R₂<-A³; D₋₁<-0; i<-M−1; s<-1

As long as i≧0, repeat the block Γ(k,s): k<-(/s)×(k+1)+s×(D _(i)+2×(D _(i) AND D _(i−1))) s<-/((k mod 2) ⊕ (k div 4))

-   -   γ(k,s): R₀<-R₀×R_(s×(k div 2))         -   i<-i−s×(k mod 2+1)

Return R₀.

Algorithm 4

(Protected Algorithm (M, M³), Version 1)

EXAMPLE 5

As seen in the context of Examples 1 and 2, for one and the same algorithm it is possible to choose between several common elementary blocks γ(k) or γ(k,s).

In the case of the (M, M³) algorithm for example, it is also possible to break down the blocks Π₀, Π₁, Π₂ in the following manner: TABLE 5 k s Π₀ (D_(i) = 0) γ0: R₀ <- R₀ × R₀; i <- i − 1 0 1 Π₁ (D_(i) = 1) and (D_(i−1) = 0) γ1: R₀ <- R₀ × R₀; i <- i − 0 0 0 γ2: R₀ <- R₀ × R₁; i <- i − 1 1 1 Π_(s) (D_(i) = 1) and (D_(i−1) = 1) γ3: R₀ <- R₀ × R₀; i <- i − 0 0 0 γ4: R₀ <- R₀ × R₀; i <- i − 0 1 0 γ5: R₀ <- R₀ × R₂; i <- i − 2 2 1

Compared with Table 4, only the values of k have been modified.

Table 5 makes it possible to calculate, as before, the value of the pointer k as a function of s and D_(i), the value of the pointer s as a function of k, and the value by which the index i must be decremented. Moreover, it is possible for example to choose the following common elementary block γ(k,s):

-   -   γ(k,s): R₀<-R₀×R_(k×s)         -   i<-i−k×s−(/Di)

Finally, a protected Algorithm 5 is derived therefrom:

Initialisation: R₀<-1; R₁<-A; R₂<-A³; D⁻¹<-0; i<-M−1; s<-1

As long as i≧0, repeat: k<-(/s)×(k+1) s<-s ⊕ D _(i) ⊕ ((D _(i-1) AND (k mod 2))

-   -   Γ(k,s): R₀<-R₀×R_(k×s)         -   i<-i−k×s−(/D_(i))

Return R₀.

Algorithm 5

(Protected Algorithm (M, M³), Version 2)

As has been seen in the above examples, it is fairly simple to obtain, in the context of the invention, a breakdown of each block Π_(j) of instructions into elementary blocks γ0, γ1, . . . , γL_(j).

However, the relationships linking the loop pointer k and the state pointer s to the variable D_(i) and/or to the variable j indexing the various blocks Π₀, Π_(j), . . . , Π_(N) become complex when the algorithm which it is sought to protect is itself complex (that is to say when it uses a large number of different blocks Π_(j), when each block Π_(j) is broken down into a large number of elementary blocks Π, etc). For certain particularly complex algorithms such as cryptographic algorithms on elliptic curves, this difficulty can even prove to be great or even insurmountable.

In order to resolve or get around this difficulty, according to another embodiment of the invention, the links between the values of the loop pointer k, the state pointer s, the index of the registers used, the index i of the variable D and the index j of the blocks Π_(j), are expressed in the form of a Table U with several inputs, as will be seen in the examples below.

In the practical implementation of the invention, the so-called Table U can for example be stored in a memory, erasable or not, of the device used. The updating of the pointers will then be effected by a reading in the memory of one or more values in the matrix U.

EXAMPLE 6

The breakdown of the “Square and Multiply” algorithm into elementary blocks is considered once again: TABLE 6 = TABLE 2 k s (D_(i) = 1) γ0: R₀ <- R₀ × R₀; i <- i − 0 0 0 γ1: R₀ <- R₀ × R₁; i <- i − 1 1 1 (D_(i) = 0) γ2: R₀ <- R₀ × R₀; i <- i − 1 2 1

A different value of k corresponds to each line in Table 6. Each elementary block γ_(k) can be written in the following form: γ_(k) =[R _(U(k,0))<-R _(U(k,1)) ×R _(U(k,2)) ; i<-i−U(k,3)]

where U(k,1) is the element of the line k and of column 1 in the following matrix: ${\left( U_{k,1} \right)\begin{matrix} {0 \leq k \leq 2} \\ {0 \leq 1 \leq 3} \end{matrix}} = \begin{pmatrix} 0000 \\ 0011 \\ 0001 \end{pmatrix}$

The matrix U is constructed in the following manner. Each row of the matrix corresponds to an elementary block γk of index k. With each column there is associated an index liable to vary from one elementary block γk to another. Here the Column 0 is associated with the index of the register in which the result of the instruction R_(α)<-R_(α)×R_(β)(α, β are equal to 0 or 1 here) is stored. Column 1 and Column 2 are associated with the indices of the registers whose product is effected by the instruction R_(α)<-R_(α)×R_(β). Finally, Column 3 is associated with the variations of the index i. The matrix U is thus obtained very simply from the table summarising the breakdown of the blocks Π_(j) into elementary blocks γk.

The constant columns of the matrix being of no interest, they can be eliminated in order to give a reduced matrix, easier to store and to use. In this way the common elementary block γ(k) is obtained: γ(k)=[R ₀<-R ₀ ×R _(U(k,0)); i<-i−U(k,1)]

with, for 0≦k ≦2 and 0≦1≦1: ${\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 2} \\ {0 \leq 1 \leq 1} \end{matrix}} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \\ 0 & 1 \end{pmatrix}$

Finally the complete protected algorithm according to the invention is derived from this.

Initialisation: R₀<-1; R₁<-A; i<-M−1; s<-1

As long as i≧0, repeat the block Γ(k,s): k<-(/s)×(k+1)+s×2×(/D _(i)) s<-U(k,1)

-   -   γ(k,s): R ₀<-R ₀ ×R _(U(k,0))         -   i<-i−s

Return R₀.

Algorithm 6

(Protected “Square and Multiply”, Version 3)

The use of a matrix is a very general method, much more general than the empirical relationships used in Examples 1 to 5 for explaining the links between the various indices used.

The expression of the links between the indices in the form of a matrix with several inputs has the advantage of being much simpler to implement and in particular being usable for all known cryptographic algorithms, including the most complex, as will be seen in a few examples of cryptographic calculation algorithms on elliptic curves (Examples 8 and 9).

EXAMPLE 7

Here the algorithm (M, M3) and its breakdown table are considered once again: TABLE 7 = TABLE 4 k s Π₀ (D_(i) = 0) γ0: R₀ <- R₀ × R₀; i <- i − 1 0 1 Π₁ (D_(i) = 1) and (D_(i−1) = 0) γ1: R₀ <- R₀ × R₀; i <- i − 0 1 0 γ2: R₀ <- R₀ × R₁; i <- i − 1 2 1 Π_(s) (D_(i) = 1) and (D_(i−1) = 1) γ3: R₀ <- R₀ × R₀; i <- i − 0 3 0 γ4: R₀ <- R₀ × R₀; i <- i − 0 4 0 γ5: R₀ <- R₀ × R₂; i <- i − 2 5 1

From Table 7, the following matrix is easily derived: ${\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 5} \\ {0 \leq 1 \leq 2} \end{matrix}} = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 2 & 1 \end{pmatrix}$

one possible expression of a common elementary block γ(k): γ(k)=[R ₀<-R ₀ ×R _(U(k,0)) ; i<-i−R _(U(k,1))]

and a protected algorithm using the common elementary block γ(k):

Initialisation: R₀<-1; R₁<-A; R₂<-A³; i<-M−1; s<-1

As long as i≧0, repeat the common block Γ(k,s): k<-(/s)×(k+1)+s×(D _(i)+2×(/D _(i) AND D _(i-1))); s<-U(k,2)

-   -   γ(k,s): R₀<-R₀×R_(U(k,0));         -   i<-i−U(k,1)

Return R₀.

Algorithm 7

(Protected Algorithm (M, M³), Version 3)

EXAMPLE 8

A cryptographic calculation algorithm on a non-supersingular elliptic curve E defined on a binary field F₂q by the following Weierstrass equation: E/F ₂ q: Y ² +X×Y=X ³ +a×X ² +b   (EQ d)

where X, Y are the affine coordinates of a point P on the curve E.

The basic operations of a cryptographic algorithm on elliptic curves are the operations of doubling of points and the operations of addition of two distinct points.

The operation of doubling of a point is defined by: P3(X3, Y3)=2×P1(X1, Y1) with X3=a+λ ²+λ Y3=(X1+X3)×λ+X3+Y1 and λ=X1+(Y1/X1)

The operation of addition of two distinct points is defined by: P(X3, Y3)=P1(X1, Y1)+P2(X2, Y2) X3=a+λ ² +λ+X1+X2 Y3=(X1+X3)×λ+X3+Y1 and λ=(Y1+Y2)/(X1+X2)

In Table 8, the operation of doubling of points and the operation of addition of two distinct points have been broken down in the form each of an equivalent elementary block γ0, γ1 (the same operations are used, possibly on different registers): TABLE 8 k s γ0: R₁ <- R₁ + R₃; R₂ <- R₂ + R₄; R₅ <- R₂/R₁; 0 1 R₁ <- R₁ + R₅; R₆ <- R₅ ²; R₆ <- R₆ + a; R₁ <- R₁ + R₆; R₂ <- R₁ + R4; R₆ <- R₁ + R₃; R₅ <- R₅ × R6; R₂ <- R₂ + R₅ γ1: R₆ <- R₁ + R₃; R₆ <- R₆ + R₃; R₅ <- R₂/R₁; 1 1 R₅ <- R₁ + R₅; R₁ <- R₅ ²; R₁ <- R₁ + a; R₁ <- R₁ + R₅; R₂ <- R₁ + R₂; R₆ <- R₁ + R₆; R₅ <- R₅ × R₆; R₂ <- R₂ + R₅

From Table 8, the following matrix is derived: ${\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 1} \\ {0 \leq 1 \leq 7} \end{matrix}} = \begin{pmatrix} 1 & 2 & 4 & 1 & 6 & 6 & 4 & 3 \\ 6 & 6 & 3 & 5 & 1 & 5 & 2 & 6 \end{pmatrix}$

The matrix comprises only two rows since only two different elementary blocks are used. The matrix comprises 8 columns, each associated with a register index varying from one row to another. Column 0 is thus associated with the index of the register (R1 or R6) in which the result of the first operation (R1+R3) is stored, Column 1 is associated with the index of the register (R2 or R6) in which the result of the second operation (R₂+R₄ or R₆+R₃) is stored, Columns 1 and 2 are associated with the registers whose contents are added during the second operation (R₂+R₄ or R₆+R₃), etc.

The matrix is to be used with the following common elementary block:

γ(k): R_(U(k,0))<-R₁+R₃; R_(U(k,1))<-R_(U(k,1))+R_(U(k,2));

-   -   R₅<-R₂/R₁; R_(U(k,3))<-R₁+R₅;     -   R_(U(k,4))<-R₅ ²;     -   R_(U(k,4))<-R_(U(k,4))+a; R₁<-R₁+R_(U(k,5));     -   R₂<-R₁+R_(U(k,6)); R₆<-R₁+R_(U(k,7));     -   R₅<-R₅·R₆; R₂<-R₂+R₅

in order to effect a protected algorithm using the common block Γ(k) in a loop of the “repeat as long as” type and performing a complex operation using basic operations (doubling of points and/or addition of points)

Initialisation: R₁<-X₁; R₂<-Y₁; R₃<-X₁; R₄<-Y₁; i<m−2; s<-1; k<-0;

As long as i≧0, repeat Γ(k,s): γ(k) s<-k−D _(i)+1 k<-(k+1)×(/s); i<-i−s;

Return (R1, R2)

Algorithm 8

(Protected Algorithm on Elliptic Curve) 

1. (canceled)
 2. A method according to claim 21, wherein the predefined number is variable from one predefined block of instructions to another.
 3. A method according to claim 21, wherein the common set of instructions comprises at least one calculation instruction that is equivalent, to a calculation instruction of each predefined block in the context of a covert channel attack.
 4. A method according to claim 3, in which the common set of instructions also comprises an instruction to update a loop pointer indicating a number of executions already performed with the common set of instructions.
 5. A method according to claim 3 wherein the common set of instructions also comprises an instruction to update a state pointer indicating whether the predefined number has been reached.
 6. A method according to claim 4 wherein the value of the loop pointer is a function of the value of the input variable and/or of the number of instructions in the selected block of instructions.
 7. A method according to claim 21, wherein, in order to successively effect several blocks of instructions chosen from amongst the plural predefined blocks of instructions each selected block of instructions is selected as a function of an input variable associated with an input index, and the common set of instructions is executed a total number of times, equal to a sum of the predefined numbers associated with each selected block of instructions.
 8. A method according to claim 7, wherein one and the same block of instructions is selected several times according to the input variable associated with the input index.
 9. A method according to claim 7, wherein at least two of the following data items, (a) the value of a loop pointer, (b) the value of a state pointer, (c) the value of the input variable, and (d) the number of instructions of the selected block of instructions, are linked by one or more mathematical functions.
 10. A method according to claim 9, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following steps: Initialisation: R₀<-1; R₁<-A; i<-M−1 As long as i>0, repeat the common set of instructions: k<-(/s)×(k+1)+s×2×(/D _(i)) s<-(k mod 2)+(k div 2) γ(k,s): R₀<-R₀×R_(k) mod 2 i<-i−s Return R₀, where R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 11. A method according to claim 9, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following steps: Initialisation: R₀<-1; R₁<-A; i<-M−1; k<-1 As long as i≧0, repeat the common set of instructions: k<-(Di) AND (/k) γ′(s,k): R₀<-R₀×R_(k) i<-i−(/k) Return R₀, where R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 12. A method according to claim 9, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following steps: Initialisation: R₀<-1; R₁<-A; i<-0; k<-1 As long as i≦M−1, repeat the common set of instructions: k<-k ⊕ D_(i) γ(k): R_(k)<-R_(k)×R₁ i<-i+k Return R₀, where R₀ and R₁ are values stored in two registers, respectively, and k is a loop pointer indicating a number of executions performed with the common set of instructions.
 13. A method according to claim 9, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following steps: Initialisation: R₀<-1; R₁<-A,; R₂<-A³; D₋₁<-0; i<-M−1; s<-1 As long as i≧0, repeat the common set of instructions: k<-(/s)×(k+1)+s×(D _(i)+2×(D _(i) AND D _(i-1))) s<-/((k mod 2) ⊕ (D(k div 4)) γ(k,s): R₀<-R₀×R_(s×(k div 2)) i<-i−s×(k mod 2+1) Return R₀, where R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 14. A method according to claim 9, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following steps: Initialisation: R₀<-1; R₁<-A; R₂<-A³; D₋₁<-0; i<-M−1; s<-1 As long as i≧0, repeat: k<-(/s)×(k+1) s<-s ⊕ D _(i) ⊕ ((D _(i-1) AND (k mod 2)) R₀<-R₀×R_(k×s) i<-i−k×s−(/D_(i)) Return R₀, where R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 15. A method according to claim 7, wherein at least two of the following data items, (a) the value of a loop pointer, (b) the value of a state pointer, (c) the value of the input variable, and (d) the number of instructions of the selected block of instructions, are linked and such linking is defined by a table with several inputs.
 16. A method according to claim 15, used in the implementation of an exponentiation calculation of the type B=A^(D), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following step: As long as i≧0, repeat the common set of instructions: k<-(/s)×(k+1)+s×2×(/D _(i)) s<-U(k,1) γ(k,s): R₀<-R₀×R_(U(k,0)) i<-i−s where (U(k,1)) is the following matrix: ${{\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 2} \\ {0 \leq 1 \leq 1} \end{matrix}} = \begin{pmatrix} 0 & 0 \\ 1 & 1 \\ 0 & 1 \end{pmatrix}},$ R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 17. A method according to claim 15, used in the implementation of an exponentiation calculation of the type B=A^(D) according to the algorithm (M, M³), with D being an integer number of M bits, and each bit (D_(i)) of D corresponding to an input variable of input index i, comprising the following step: As long as i≧0, repeat the common set of instructions: k<-(/s)×(k+1)+s×(D _(i)+2×(/D _(i) AND D _(i-1))) s<-U(k,2) γ(k,s): R₀<-R₀×R_(U(k,0)); i<-i−U(k,1) where (U(k,1)) is the following matrix: ${{\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 5} \\ {0 \leq 1 \leq 2} \end{matrix}} = \begin{pmatrix} 0 & 1 & 1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 2 & 2 & 1 \end{pmatrix}},$ R₀ and R₁ are values stored in two registers, respectively, k is a loon pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 18. A method according to claim 15, used in the implementation of a calculation on an elliptic curve in affine coordinates, a calculation using operations of the addition or doubling of points type, and in which the following step is performed: As long as i≧0, repeat: γ(k): R_(U(k,0))<-R₁+R₃; R_(U(k,1))<-R_(U(k,1))+R_(U(k,2)); R₅<-R₂/R₁; R_(U(k,3))<-R₁+R₅; R_(U(k,4))<-R₅ ²; R_(U(k,4))<-R_(U(k,4))+a; R₁<-R₁+R_(U(k,5)); R₂<-R₁+R_(U(k,6)); R₆<-R₁+R_(U(k,7)); R₅<-R₅·R₆; R₂<-R₂+R₅ s<-k−D _(i)+1 k<-(k+1)×(/s); i<-i−s; where (U(k,1)) is the following matrix: ${\left( {U\left( {k,1} \right)} \right)\quad\begin{matrix} {0 \leq k \leq 1} \\ {0 \leq 1 \leq 10} \end{matrix}} = {{\begin{pmatrix} 1 & 2 & 4 & 1 & 6 & 6 & 4 & 3 \\ 6 & 6 & 3 & 5 & 1 & 5 & 2 & 6 \end{pmatrix}\left\lbrack \left\lbrack . \right\rbrack \right\rbrack}\quad\bot}$ R₀ and R₁ are values stored in two registers, respectively, k is a loop pointer indicating a number of executions performed with the common set of instructions, and s is a state pointer indicating whether the predefined number has been reached.
 19. A method for obtaining an elementary set of instructions common to a plurality of predefined blocks of instructions, for implementing a cryptographic calculation method according to claim 21, comprising the following steps: E1: breaking down each predefined block of instructions into a series of elementary blocks that are equivalent in the context of a covert channel attack, and classifying all the elementary blocks, E2: identifying a common elementary block that is equivalent to all the elementary blocks of all the predefined blocks of instructions, E3: identifying a common block comprising at least the common elementary block previously identified and an instruction to update a loop pointer such that an execution of the common elementary block associated with the value of the loop pointer and an execution of the elementary block with a rank equal to the value of the loop pointer are identical.
 20. A method according to claim 19, wherein, during step E1, at least one fictional instruction is added to at least one predefined block of instructions.
 21. A method for implementing a cryptographic calculation in an electronic device, comprising the following steps: selecting a block of instructions from amongst a plurality of predefined blocks of instructions, as a function of an input variable; and executing a set of instructions that is common to the plurality of predefined blocks of instructions a predefined number of times, wherein said predefined number is associated with the selected block of instructions.
 22. A method according to claim 5, wherein the value of the state pointer is a function of the value of the input variable and/or of the number of instructions in the selected block of instructions.
 23. A method according to claim 15, wherein said several inputs comprise a matrix.
 24. A method according to claim 21, wherein said electronic device is a chip card. 